초록 열기/닫기 버튼
Let X be a locally compact Hausdorff space and C1 (X ) be the ring of all real valued continuous functions on X which vanish at infinity. Suppose CK (X ) is the set of all real valued continuous functions on X which have compact support, then CK (X ) is a subring of C1 (X ). We have shown that every maximal ideal of C1 (X ) is fixed from which it follows that if CK (X ) 6= C1 (X ) then CK (X ) is a proper ideal of C1 (X ) which is not extendable to any maximal ideal. By using the same result we have further established that two, locally compact Hausdorff spaces X and Y are homeomorphic if and only if C1 (X ) and C1 (Y) (respectivelyCK (X ) and CK (Y)) are isomorphic.
키워드열기/닫기 버튼
locally compact spaces, rings of continuous functions with compact support, rings of continuous functions which vanish at infinity, maximal ideals, structure spaces
